# Betatron motion with coupling of horizontal and vertical degrees of freedom

## Abstract

The Courant-Snyder parameterization of one-dimensional linear betatron motion is generalized to two-dimensional coupled linear motion. To represent the 4 x 4 symplectic transfer matrix the following ten parameters were chosen: four beta-functions, four alpha-functions and two betatron phase advances which have a meaning similar to the Courant-Snyder parameterization. Such a parameterization works equally well for weak and strong coupling and can be useful for analysis of coupled betatron motion in circular accelerators as well as in transfer lines. Similarly, the transfer matrix, the bilinear form describing the phase space ellipsoid and the second order moments are related to the eigen-vectors. Corresponding equations can be useful in interpreting tracking results and experimental data.

- Authors:

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab., Batavia, IL (US)

- Sponsoring Org.:
- USDOE Office of Energy Research (ER) (US)

- OSTI Identifier:
- 804898

- Report Number(s):
- FERMILAB-Conf-01/439

TRN: US0205603

- DOE Contract Number:
- AC02-76CH03000

- Resource Type:
- Conference

- Resource Relation:
- Conference: Snowmass 2001, Snowmass, CO (US), 06/30/2001--07/21/2001; Other Information: PBD: 21 Nov 2002

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; ACCELERATORS; BETATRONS; DEGREES OF FREEDOM; PHASE SPACE

### Citation Formats

```
Bogacz, S A, and Lebedev, V A.
```*Betatron motion with coupling of horizontal and vertical degrees of freedom*. United States: N. p., 2002.
Web.

```
Bogacz, S A, & Lebedev, V A.
```*Betatron motion with coupling of horizontal and vertical degrees of freedom*. United States.

```
Bogacz, S A, and Lebedev, V A. Thu .
"Betatron motion with coupling of horizontal and vertical degrees of freedom". United States. https://www.osti.gov/servlets/purl/804898.
```

```
@article{osti_804898,
```

title = {Betatron motion with coupling of horizontal and vertical degrees of freedom},

author = {Bogacz, S A and Lebedev, V A},

abstractNote = {The Courant-Snyder parameterization of one-dimensional linear betatron motion is generalized to two-dimensional coupled linear motion. To represent the 4 x 4 symplectic transfer matrix the following ten parameters were chosen: four beta-functions, four alpha-functions and two betatron phase advances which have a meaning similar to the Courant-Snyder parameterization. Such a parameterization works equally well for weak and strong coupling and can be useful for analysis of coupled betatron motion in circular accelerators as well as in transfer lines. Similarly, the transfer matrix, the bilinear form describing the phase space ellipsoid and the second order moments are related to the eigen-vectors. Corresponding equations can be useful in interpreting tracking results and experimental data.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2002},

month = {11}

}